Odd-order cyclic group is characteristic in holomorph

This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., odd-order cyclic group) must also satisfy the second group property (i.e., holomorph-characteristic group)
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Statement

Any Odd-order cyclic group (?) is a Characteristic subgroup (?) inside its holomorph.

Related facts

Related facts about odd-order cyclic groups

• Odd-order cyclic group equals commutator subgroup of holomorph
• Odd-order cyclic group is fully invariant in holomorph
• Odd-order abelian group not is fully invariant in holomorph

Breakdown for other cyclic groups

• Cyclic group not is characteristic in holomorph
• Cyclic group not is fully invariant in holomorph

Other related facts

• Additive group of a field implies characteristic in holomorph, because additive group of a field is monolith in holomorph (in particular, any elementary abelian group is characteristic in its holomorph).
• Odd-order elementary abelian group is fully invariant in holomorph
• Odd-order abelian group not is fully invariant in holomorph

Facts used

1. Odd-order cyclic group equals commutator subgroup of holomorph
2. Commutator subgroup is characteristic

Proof

The proof follows directly from facts (1) and (2).